5  Population Variances (8.3.1 - 8.3.2)

Confidence intervals relating to the population variances require that the distribution a sample is taken from is a normal distribution. It is therefore always important to check that the sample data reasonably follow a normal distribution. If this is not the case, then any confidence intervals calculated using the formulae stated below will be inaccurate.

There are ways to calculate confidence intervals relating to population variances when this assumption of normally distributed data is not valid, however these are not covered in S2S.

5.1 Sampling from a Normal population (8.3.1)

Suppose a sample of size \(n\) is taken from a normal population, \(N(\mu, \sigma)\). A \((1-\alpha)\cdot100\%\) confidence interval for the population variance, \(\sigma^2\), is given by,

\[\begin{equation} \tag{8.26} CI_{1-\alpha}(\sigma^2)=\left[\frac{(n-1)s^2}{\chi^2_{1-\frac{\alpha}{2};n-1}},\frac{(n-1)s^2}{\chi^2_{\frac{\alpha}{2};n-1}}\right] \end{equation}\]

• \(s^2\) is the sample variance, which can be found using, \[s^2=\frac{\sum_{i=1}^n(x_i-\bar{x})^2}{n-1}=\frac{\sum_{i=1}^nx_i^2-n\bar x^2}{n-1}\]

Note that this confidence interval is not centred around the point estimate for the population variance, which is the sample variance, \(s^2\). This is because the \(\chi^2\) distribution is not symmetric.

To find the corresponding quantiles from a \(\chi^2\) distribution with \(n-1\) degrees of freedom, \(\chi^2_{1-\frac{\alpha}{2};n-1}\) and \(\chi^2_{\frac{\alpha}{2};n-1}\), the function qchisq() can be used (see the Probability functions section of Lab 3 for more details).

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See Section 8.3.1 of Probability and Statistics with R to see the derivation of this result and some examples of using it in practise.

5.2 Ratio of population variances (8.3.2)

When two samples of size \(n_X\) and \(n_Y\) are taken from independent normal distributions, \(N(\mu_X,\sigma_X)\) and \(N(\mu_Y,\sigma_Y)\), we might be interested in determining whether the variances of each distribution are equal. We do this by estimating the ratio of the two population variances, \(\frac{\sigma_X^2}{\sigma_Y^2}\), and constructing a confidence interval around the estimated ratio. If this interval contains the value 1, this is evidence that the two population variances are equal.

A \((1-\alpha)\cdot100\%\) confidence interval for the ratio of population variances, \(\frac{\sigma_X^2}{\sigma_Y^2}\), is given by,

\[\begin{equation} \tag{8.31} CI_{1-\alpha}\left(\frac{\sigma_X^2}{\sigma_Y^2}\right)=\left[f_{\frac{\alpha}{2};n_Y-1,n_X-1}\cdot\frac{s_X^2}{s_Y^2},f_{1-\frac{\alpha}{2};n_Y-1,n_X-1}\cdot\frac{s_X^2}{s_Y^2}\right] \end{equation}\]

• \(s_X^2\) is the sample variance for the first sample, which can be found using, \[s_X^2=\frac{\sum_{i=1}^{n_X}(x_i-\bar{x})^2}{n_X-1}=\frac{\sum_{i=1}^{n_X}x_i^2-n\bar x^2}{n_X-1}\]

• \(s_Y^2\) is the sample variance for the second sample, which can be found using, \[s_Y^2=\frac{\sum_{i=1}^{n_Y}(y_i-\bar{y})^2}{n_Y-1}=\frac{\sum_{i=1}^{n_Y}y_i^2-n\bar y^2}{n_Y-1}\]

Typically, the sample which has the largest sample variance will be used as the first group. That is, choose which sample is \(X\) and which is \(Y\) so that \(s_X^2>s_Y^2\).

To find the corresponding quantiles from an \(F\) distribution with \(n_Y-1\) and \(n_X-1\) degrees of freedom, \(f_{\frac{\alpha}{2};n_Y-1;n_X-1}\) and \(f_{1-\frac{\alpha}{2};n_Y-1;n_X-1}\), the function qf() can be used (see the Probability functions section of Lab 3 for more details).

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See Section 8.3.2 of Probability and Statistics with R to follow examples of using this result.